Positive solutions for a class of nonlinear Hadamard fractional differential equations with a parameter

Haisong Huang,Weihua Liu

doi:10.1186/s13662-018-1551-9

Abstract

In this paper, we investigate a class of boundary value problem of nonlinear Hadamard fractional differential equations with a parameter. By means of the properties of the Green function and Guo–Krasnosel’skii fixed-point theorem on cones, the existence and nonexistence of positive solutions are obtained. Finally, some examples are presented to show the effectiveness of our main results.

Highlights

Fractional differential equations have given rise to abroad attention of many researchers by the intensive development of the theory of fractional calculus itself
The existence and multiplicity of solutions or positive solutions for nonlinear boundary value problems involving fractional differential equations with kinds of boundary value conditions were studied by some wellknown fixed-point theorems, the lower and upper solutions method and the monotone iterative technique; see [13, 14] and the references therein
Henderson and Luca investigated the positive solutions of nonlinear boundary value problems for systems of fractional differential equations in the book [17]

Summary

Introduction

Fractional differential equations have given rise to abroad attention of many researchers by the intensive development of the theory of fractional calculus itself. By employing the Avery–Henderson fixed-point theorem, Li [11] obtained the existence of positive solutions as considered for a fractional differential equation with p-Laplacian operator. In [12], existence and uniqueness results for a new class of boundary value problems of sequential fractional differential equations with nonlocal non-separated boundary conditions involving lower-order fractional derivatives were given by some standard fixed-point theorems. The existence and multiplicity of solutions or positive solutions for nonlinear boundary value problems involving fractional differential equations with kinds of boundary value conditions were studied by some wellknown fixed-point theorems, the lower and upper solutions method and the monotone iterative technique; see [13, 14] and the references therein. In [18], by applying the fixed-point theorem due to Leggett–Williams, the authors considered the

Objectives

Conclusion